Theoretcal Economcs 5 (00), 445 478 555-756/000445 Unform topologes on types Y-Chun Chen Department of Economcs, Natonal Unversty of Sngapore Alfredo D Tllo IGIER and Department of Economcs, Unverstà Lug Boccon Eduardo Fangold Department of Economcs, Yale Unversty Syang Xong Department of Economcs, Rce Unversty We study the robustness of nterm correlated ratonalzablty to perturbatons of hgher-order belefs. We ntroduce a new metrc topology on the unversal type space,called unform-weak topology, under whch two types are close f they have smlar frst-order belefs, attach smlar probabltes to other players havng smlar frst-order belefs, and so on, where the degree of smlarty s unform over the levels of the belef herarchy. Ths topology generalzes the now classc noton of proxmty to common knowledge based on common p-belefs (Monderer and Samet 989). We show that convergence n the unform-weak topology mples convergence n the unform-strategc topology (Dekel et al. 006). Moreover, when the lmt s a fnte type, unform-weak convergence s also a necessary condton for convergence n the strategc topology. Fnally, we show that the set of fnte types s nowhere dense under the unform strategc topology. Thus, our results shed lght on the connecton between smlarty of belefs and smlarty of behavors n games. Keywords. Ratonalzablty, ncomplete nformaton, hgher-order belefs, strategc topology, Electronc Mal game. JEL classfcaton. C70, C7. Y-Chun Chen: ecsycc@nus.edu.sg Alfredo D Tllo: alfredo.dtllo@unboccon.t Eduardo Fangold: eduardo.fangold@yale.edu Syang Xong: xong@rce.edu We are very grateful to a co-edtor and three anonymous referees for ther comments and suggestons, whch greatly mproved ths paper. We also thank Perpaolo Battgall, Martn Crpps, Edde Dekel, Jeffrey C. Ely, Amanda Fredenberg, Drew Fudenberg, Qngmn Lu, George J. Malath, Stephen Morrs, Marcn Pesk, Dov Samet, Marcano Snscalch, Aaron Sojourner, Tomasz Strzaleck, Satoru Takahash, Jonathan Wensten, and Muhamet Yldz for ther nsghtful comments. Chen and Xong gratefully acknowledge fnancal support from the NSF (Grant SES 080333) and the Northwestern Unversty Economc Theory Center. Copyrght 00 Y-Chun Chen, Alfredo D Tllo, Eduardo Fangold, and Syang Xong. Lcensed under the Creatve Commons Attrbuton-NonCommercal Lcense 3.0. Avalable at http://econtheory.org. DOI: 0.398/TE46
446 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00). Introducton The Bayesan analyss of ncomplete nformaton games requres the specfcaton of a type space, whch s a representaton of the players uncertanty about fundamentals, ther uncertanty about the other players uncertanty about fundamentals, and so on, ad nfntum. Thus the strategc outcomes of a Bayesan game may depend on entre nfnte herarches of belefs. Crtcally, n some games ths dependence can be very senstve at the tals of the herarches, so that a mspecfcaton of hgher-order belefs, even at arbtrarly hgh orders, can have a large mpact on the predctons of strategc behavor, as shown by the Electronc Mal game of Rubnsten (989). As a matter of fact, ths phenomenon s not specal to the E-Mal game. Recently, Wensten and Yldz (007) have shown that n any game satsfyng a certan payoff rchness condton, f a player has multple actons that are consstent wth nterm correlated ratonalzablty the soluton concept that embodes common knowledge of ratonalty then any of these actons can be made unquely ratonalzable by sutably perturbng the player s hgher-order belefs at any arbtrarly hgh order. Ths phenomenon rases a conceptual ssue: f predctons of strategc behavor are not robust to mspecfcaton of hgherorder belefs, then the common practce n appled analyss of modelng uncertanty usng small type spaces often fnte may gve rse to spurous predctons. A natural approach to study ths robustness problem s topologcal. Consder the correspondence that maps each type of player nto hs set of nterm correlated ratonalzable (ICR) actons. The fraglty of strategc behavor dentfed by Rubnsten (989) and Wensten and Yldz (007) can be recast as a certan knd of dscontnuty of the ICR correspondence n the product topology over herarches of belefs,.e., the topology of weak convergence of k-order belefs, for each k. Whle n every game the ICR correspondence s upper hemcontnuous n the product topology, lower hemcontnuty can fal even for the strct ICR correspondence a refnement of ICR that requres the ncentve constrants to hold wth strct nequalty. Strctness rules out ncentves that hnge on a knfe edge, whch can always be destroyed by sutably perturbng the payoffs of the game. Indeed, nonstrct soluton concepts are known to fal lower hemcontnuty n other contexts, e.g., n complete nformaton games, Nash equlbrum, and, n fact, even best-reply correspondences fal to be lower hemcontnuous wth respect to payoff perturbatons. By contrast, the strct Nash equlbrum and the strct best-reply correspondences are lower hemcontnuous. It s, therefore, surprsng that ths form of contnuty breaks down when t comes to perturbatons of hgher-order belefs. There exst, of course, fner topologes under whch the ICR correspondence s upper hemcontnuous and the strct ICR correspondence s lower hemcontnuous n all games. The coarsest such topology s the strategc topology ntroduced by Dekel et al. (006); t embodes the mnmum restrctons on the class of admssble perturbatons of hgher-order belefs necessary to render ratonalzable behavor contnuous. Thus See Dekel et al. (007, Proposton ) and Battgall et al. (008, Theorem4). Here, the noton of strctness s actually qute strong: the slack n the ncentve constrants s requred to be bounded away from zero unformly on a best-reply set. Despte ths, the strct ICR correspondence fals to be lower hemcontnuous n the product topology.
Theoretcal Economcs 5 (00) Unform topologes on types 447 the strategc topology gves a tght measure of the robustness of strategc behavor: f the analyst consders any larger set of perturbatons, he s bound to make a nonrobust predcton n some game. Gven ths sgnfcance, we beleve the strategc topology deserves closer examnaton. Indeed, Dekel et al. (006) only defne t mplctly n terms of proxmty of behavor n games, as opposed to explctly usng some noton of proxmty of probablty measures. Ths leaves open the mportant queston as to what proxmty n the strategc topology means n terms of the belefs of the players. To address ths queston, we ntroduce a new metrc topology on types, called unform-weak topology, under whch a sequence of types (t n ) n converges to a type t f the k-order belef of t n weakly converges to that of t and the rate of convergence s unform over k. More precsely, for each k, we consder the Prohorov metrc, d k,overk-order belefs a standard metrc that metrzes the topology of weak convergence of probablty measures and then defne the unform-weak topology as the topology of convergence n the metrc d UW sup k d k. Our frst man result, Theorem, s that convergence n the unform-weak topology mples convergence n the unform-strategc topology. The latter, also ntroduced by Dekel et al. (006), s the coarsest topology on types under whch the ICR correspondence s upper hemcontnuous and the strct ICR correspondence s lower hemcontnuous, where the contnuty s now requred to hold unformly across all games. 3 In partcular, Theorem mples that convergence n the unform-weak topology s a suffcent condton for convergence n the strategc topology. The unform-weak topology s nterestng n ts own rght, as t generalzes the classc noton of approxmate common knowledge due to Monderer and Samet (989). Gven a payoff-relevant parameter θ, say that a type of a player has common p-belef n θ f he assgns probablty no smaller than p to θ, assgns probablty no smaller than p to the event that θ obtans and the other players assgn probablty no smaller than p to θ,and so forth, ad nfntum. A sequence of types (t n ) n has asymptotc common certanty of θ f for every p<, t n has common p-belef n θ for all n large enough. Monderer and Samet (989) use ths noton of proxmty to common knowledge to study the robustness of Nash equlbrum to small amounts of ncomplete nformaton. Although they focus on an ex ante noton of robustness and consder only common pror perturbatons, ther man result has the followng counterpart n our nterm, noncommon pror, nonequlbrum framework. If a sequence of types (t n ) n has asymptotc common certanty of θ, then, for every game, every acton that s strctly nterm correlated ratonalzable when θ s common certanty remans nterm correlated ratonalzable for type t n, for all n large enough. It turns out that asymptotc common certanty of θ s equvalent to unform-weak convergence to the type that has common certanty of θ (.e., common -belef). Thus, our Theorem s a generalzaton of Monderer and Samet s (989) man result to envronments where the lmt game has ncomplete nformaton. 3 See Secton 3 for the precse defnton of the modulus of contnuty on whch the unformty s based.
448 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) An mportant corollary of Theorem s that the strategc, unform-strategc, and product topologes generate the same σ-algebra. 4 Indeed, a fundamental result of Mertens and Zamr (985), whch s the Bayesan foundaton of Harsany s (967 968) model of types, s that the space of herarches of belefs, called the unversal type space, exhausts all the relevant uncertanty of the players when endowed wth the product σ- algebra. It s reassurng to know that ths unversalty property remans vald when the players can reason about any strategc event. 5 Our second man result, Theorem, s that unform-weak convergence s also a necessary condton for strategc convergence when the lmt s a fnte type,.e., a type belongng to a fnte type space. Indeed, for any fnte type t and for any sequence of (possbly nfnte) types (t n ) n that fals to converge to t unform-weakly, we construct a game n whch an acton s strctly nterm correlated ratonalzable for t, but not nterm correlated ratonalzable for t n, nfntely often along the sequence. 6 Thus, the unform-weak topology fully characterzes the strategc topology around fnte types. Moreover, the assumpton that the lmt s a fnte type cannot be dspensed wth. Under the unform-weak topology, the unversal type space s not separable,.e.,t does not contan a countable dense subset; by contrast, Dekel et al. (006) show that a countable set of fnte types s dense under the strategc topology. 7 Ths mples the exstence of nfnte types to whch unform-weak convergence s not a necessary condton for strategc convergence. (We explctly construct such an example n Secton 4.) Whle ths fact mposes a natural lmt to our analyss, fnte type spaces play a promnent role n both appled and theoretcal work, so t s mportant to know that our suffcent condton for strategc convergence s also necessary n ths case. Fnte types are also the focus of our thrd man result, Theorem 3. We show that, under the unform-strategc topology, the set of fnte types s nowhere dense,.e.,ts closure has an empty nteror. To understand the conceptual mplcatons of ths result, recall that Dekel et al. (006) demonstrate the denseness of fnte types under the nonunform verson of the strategc topology. 8 Arguably, ths result provdes a compellng justfcaton for why t mght be wthout loss of generalty to model uncertanty wth fnte type spaces: Irrespectve of how large the true type space T s, for any gven game there s always a fnte type space T wth the property that the predctons of strategc behavor 4 Ths s because unform-weak balls are countable ntersectons of fnte-order cylnders and the strategc topologes are sandwched between the unform-weak and the product topologes, by Theorem. 5 Morrs (00, Secton 4.) rases the queston of whether the Mertens Zamr constructon s stll meanngful when strategc topologes are assumed. 6 Ths complements the man result of Wensten and Yldz (007), who fx a game (satsfyng a payoffrchnessassumpton) andafntetype t, andthen construct a sequence of types convergng to t n the product topology such that the behavor of t s bounded away from the behavor of all types n the sequence. By way of contrast, we fx a sequence of types thatfalstoconvergetoafntetypet n the unform-weak topology and then construct a game for whch the behavor of t s bounded away from the behavor of the types n the sequence nfntely often. 7 Whle Dekel et al. (006) state only the weaker result that the set of all fnte types s dense n the strategc topology, ther proof actually establshes the stronger result above. 8 Mertens and Zamr (985) prove the denseness of fnte types under the product topology. Dekel et al. (006) argue that ths result does not provde a sound justfcaton for restrctng attenton to fnte types, for strategc behavor s not contnuous n the product topology.
Theoretcal Economcs 5 (00) Unform topologes on types 449 based on T are arbtrarly close to those based on T. Our nowhere denseness result mples that such fnte type space T cannot be chosen ndependently of the game. Ths s partcularly relevant for envronments such as those of mechansm desgn, where the game both payoffs and acton sets s not a pror fxed. More generally, our result mples that the unform-strategc topology s strctly fner than the strategc topology. Thus, whle a pror these two notons of strategc contnuty seem equally compellng, assumng one or the other can have a large mpact on the ensung theory. The exercse n ths paper s smlar n sprt to that of Monderer and Samet (996) and Kaj and Morrs (998), who, lke us, consder perturbatons of ncomplete nformaton games. These papers provde belef-based characterzatons of strategc topologes for Bayesan Nash equlbrum n countable partton models à la Aumann (976). However, snce both of these papers assume a common pror and adopt an ex ante approach, whle we adopt an nterm approach wthout mposng a common pror, t s dffcult to establsh a precse connecton. 9 Another mportant dfference between ther approachandourssnthedstnctpayoff-relevance constrants adopted: we fx the set of payoff-relevant states, so our games cannot have payoffs that depend drectly on players hgher-order belefs; Monderer and Samet (996) andkaj and Morrs (998) have no such payoff-relevance constrant. The connecton between unform and strategc topologes frst appears n Morrs (00), who studes a specal class of games, called hgher-order expectaton (HOE), games, and shows that the topology of unform convergence of hgher-order terated expectatons s equvalent to the coarsest topology under whch a certan noton of strct ICR correspondence dfferent from the one we consder s lower hemcontnuous n every game of the HOE class. 0 Compared to the unform-weak topology, the topology of unform convergence of terated expectatons s nether fner nor coarser, even around fnte types. We further elaborate on ths relatonshp n Secton 5. Ths paper s also related to contemporaneous work by Ely and Pęsk (008). Followng ther termnology, a type t s crtcal f, under the product topology, the strct ICR correspondence s dscontnuous at t n some game. Ely and Pęsk (008) provde an nsghtful characterzaton of crtcal types n terms of a common belef property: a type s crtcal f and only f, for some p>0, t has common p-belef n some closed (n product topology) proper subset of the unversal type space. Conceptually, ths result shows that the usual type spaces that appear n applcatons consst almost entrely of crtcal types, as these type spaces typcally embody nontrval common belef assumptons. For nstance, all fnte types are crtcal and so are almost all types belongng to a common 9 Monderer and Samet (996) fx the common pror and consder proxmty of nformaton parttons, whereas Kaj and Morrs (998) vary the common pror on a fxed nformaton structure. For ths reason, the precse connecton between these papers s already unclear. 0 Morrs (00) defnes hs strategc topology for HOE games usng a dstance that makes no reference to ICR. But, as we clamed above, t can be shown that hs strategc topology concdes wth the coarsest topology under whch a certan noton of strct ICR correspondence s contnuous n every HOE game. The noton of strctness mplct n Morrs (00) analyss, unlke ours, does not requre the slack n the ncentve constrants to be unform. Moreover, they show that under the product topology the regular types,.e., those types whch are not crtcal, form a resdual subsetoftheunversaltypespace astandardtopologcalnoton ofa generc set.
450 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) pror type space. Thus ElyandPęsk s (008) resulttellsus when basedon thecommon belefs of the players there wll be some game and some product-convergent sequence along whch strategc behavor s dscontnuous, whereas we dentfy a condton for an arbtrary sequence to dsplay contnuous strategc behavor n all games. The rest of the paper s organzed as follows. Secton ntroduces the standard model of herarches of belefs and type spaces, and revews the soluton concept of ICR. Secton 3 revews the strategc and unform-strategc topologes of Dekel et al. (006), ntroduces the unform-weak topology, and presents our two man results concernng the relatonshp between these topologes (Theorems and ). Secton 4 examnes the nongenercty of fnte types under the unform-strategc and unform-weak topologes, and presents the nowhere denseness result (Theorem 3). Secton 5 dscusses the relaton wth some other topologes. Secton 6 concludes wth some open questons for future research.. Prelmnares Throughout the paper, we fx a two-player set I and a fnte set of payoff-relevant states wth at least two elements. Gven a player I, wewrte to desgnate the other player n I. All topologcal spaces, when vewed as measurable spaces, are endowed wth ther Borel σ-algebra. For a topologcal space S, wewrte (S) to desgnate the space of probablty measures over S equpped wth the topology of weak convergence. Unless explctly noted, all product spaces are endowed wth the product topology and subspaces are endowed wth the relatve topology.. Herarches of belefs and types Our formulaton of ncomplete nformaton follows Mertens and Zamr (985). 3 Defne X 0 =,andx = X 0 (X 0 ),and,foreachk, defne recursvely { } k X k = (θ μ μ k ) X 0 (X l ) : marg X l μ l = μ l l = k l= By vrtue of the above coherency condton on margnal dstrbutons, each element of X k s determned by ts frst and last coordnates, so we can dentfy X k wth (X k ). For each I and k, welett k = (X k ) desgnate the space of k-order belefs of player,sothatt k = ( T k ). ThespaceT of herarches of belefs of player s { } T = (μ k ) k (X k ) : marg X k μ k = μ k k k We restrct attenton to two-player games for ease of notaton. Our results reman vald wth any fnte number of players. 3 An alternatve, equvalent formulaton s found n Brandenburger and Dekel (993).
Theoretcal Economcs 5 (00) Unform topologes on types 45 Snce s fnte, T s a compact metrzable space. Moreover, there s a unque mappng μ : T ( T ) that s belef preservng,.e., for all t = (t t ) T and k, μ (t )θ (π k ) (E)]=t k+ θ E] for all θ and measurable E T k where π k s the natural projecton of T onto T k. Furthermore, the mappng μ s a homeomorphsm, so to save on notaton, we dentfy each herarchy of belef t T wth ts correspondng belef μ (t ) over T. Smlarly, for each t T,wewrtet k T k nstead of the more cumbersome π k(t ). Herarches of belefs can be mplctly represented usng a type space,.e., a tuple (T φ ) I,whereeachT s a Polsh space of types and each φ : T ( T ) s a measurable functon. Indeed, every type t T s mapped nto a herarchy of belefs ν (t ) = (ν k(t )) k n a natural way: ν (t ) = marg φ (t ) and, for k, ν k (t )θ E]=φ (t )θ (ν k ) (E)] for all θ and measurable E T k Thetypespace(T μ ) I s called the unversal type space, snce for every type space (T φ ) I there s a unque belef-preservng mappng from T nto T,namelythemappng ν above. 4 When the mappngs (ν ) I are njectve, the type space (T φ ) I s called nonredundant. In ths case, (ν ) I are measurable embeddngs onto ther mages (ν (T )) I, whch are measurable and can be vewed as a nonredundant type space, snce we have μ (ν (t )) ν (T )]= for all I and t T.Conversely,any(T ) I such that T T and μ (t ) T ]= for all I and t T can be vewed as a nonredundant type space.. Bayesan games and nterm correlated ratonalzablty A game s a tuple G = (A g ) I,whereA s a fnte set of actons for player and g : A A M M] s hs payoff functon, wth M>0 an arbtrary bound on payoffs that we fx throughout. 5 We wrte G to denote the set of all games and, for each nteger m, wewrteg m for the set of games wth A m for all I. The soluton concept of nterm correlated ratonalzablty (ICR) was ntroduced n Dekel et al. (007). Gven a γ R, atypespace(t φ ) I,andagameG, for each player I, nteger k 0, andtypet T,weletR k (t G γ) A desgnate the set of k-order γ-ratonalzable actons of t. These sets are defned as: R 0 (t G γ)= A and recursvely for each nteger k, R k (t G γ) s the set of all actons a A for whch there s a conjecture,.e., a measurable functon σ : T (A ) such that 6 supp σ (θ t ) R k (t G γ) (θ t ) T () 4 To say that ν s belef-preservng means that μ (ν (t ))θ E]=φ (t )θ (ν ) (E)] for all θ and measurable E T. 5 We wll also denote by g the payoff functon n the mxed extenson of G,wrtngg (α α θ)wth the obvous meanng for any α (A ) and α (A ). 6 Relaxng condton () by requrng t to hold only for φ (t )-almost every (θ t ) would not alter the defnton of ratonalzablty. Indeed, any conjecture that has a (k )-order ratonalzable support φ (t )- almost everywhere can be changed nto one that yelds the same expected payoff and satsfes the condton
45 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) and for all a A, T g (a σ (θ t ) θ) g (a σ (θ t ) θ) ] φ (t )(dθ dt ) γ () For future reference, a conjecture σ : T (A ) that satsfes the former condton wll be called a (k )-order γ-ratonalzable conjecture. Thesetofγ-ratonalzable actons of type t s then defned as R (t G γ)= k R k (t G γ) Fnally, followng Ely and Pęsk (008), an acton a A s strctly nterm correlated γ- ratonalzable for type t and we wrte a R (t G γ)f a R (t G γ ) for some γ <γ. As shown n Dekel et al. (007), R (t G γ) s nonempty for every game G, typet and γ 0. 7 Interm correlated ratonalzablty has a characterzaton n terms of best-reply sets. A par of measurable functons ς : T A, I, hastheγ-best-reply property f for each I and t T,eachactona ς (t ) s a γ-best reply for t to a conjecture σ : T (A ) wth supp σ (θ t ) ς (t ) (θ t ) T If (ς ) I has the γ-best-reply property, then ς (t ) R (t G γ)for all I and t T.As shown n Dekel et al. (007), the par (R ( G γ)) I s the maxmal par of correspondences wth the γ-best-reply property. Ths means there s no other par (ς ) I wth the γ-best-reply property such that R (t G γ) ς (t ) for each I and t T,wth strct ncluson for some I and t T. Therefore, an acton s γ-ratonalzable for a type t f and only f t s a γ-best reply to a γ-ratonalzable conjecture,.e., a conjecture σ : T (A ) such that supp σ (θ t ) R (t G γ) (θ t ) T Dekel et al. (007)also show that the set of γ-ratonalzable actons of a type s determned by the nduced herarchy of belefs. Indeed, for any k, any two types (possbly belongng to dfferent type spaces) mappng nto the same k-order belef must have the same set of k-order γ-ratonalzable actons. Ths has two mplcatons. Frst, for nterm correlated ratonalzablty, t s wthout loss of generalty to dentfy types wth ther correspondng herarches. Thus, n what follows we restrct attenton to type spaces (T ) I wth T T and t T ]= for all I and t T. 8 Accordngly, we take the unversal type space T to be the doman of the correspondence R ( G γ): T A.Second, everywhere. Ths s possble because the correspondence R k s upper hemcontnuous, and hence t admts a measurable selecton by the Kuratowsk Ryll Nardzewsk selecton theorem (see, e.g., Alprants and Border 999). 7 Note that for γ< M,wehaveR (t G γ)=, and for γ>m we have R (t G γ)= A. 8 Recall that we dentfy each type t T wth hs belef μ (t ) ( T ).
Theoretcal Economcs 5 (00) Unform topologes on types 453 to establsh whether an acton s k-order γ-ratonalzable for a type t,wecanrestrct attenton to (k )-order γ-ratonalzable conjectures σ, whch are measurable wth respect to (k )-order belefs. 9 Fnally, the followng result shows that, smlar to ratonalzablty n complete nformaton games, nterm correlated ratonalzablty has a characterzaton n terms of terated domnance, where the noton of domnance now becomes an nterm one. Proposton. Fx γ and a game G = (A g ) I.Foreachk,player I,typet T, and acton a A, we have a R k (t G γ)f and only f, for each α (A \{a }),there exsts a measurable σ : T (A ) wth such that supp σ (θ t ) R k (t G γ) (θ t ) T (3) T g (a σ (θ t ) θ) g (α σ (θ t ) θ) ] t (dθ dt ) γ The proof of ths proposton, relegated to the Appendx, uses a separaton argument analogous to that whch establshes the equvalence between strctly domnated and never best-reply strateges n complete nformaton games. Here, too, the usefulness of the result comes from the fact that to check whether an acton s ratonalzable for a type, we are able to reverse the order of quantfers and seek a possbly dfferent conjecture for each possble (mxed) devaton. 3. Topologes on types The strategc (or smply S) topology ntroduced n Dekel et al. (006) s the coarsest topology on the unversal type space T under whch the ICR correspondence s upper hemcontnuous and the strct ICR correspondence s lower hemcontnuous n all games. More explctly, followng a formulaton due to Elyand Pęsk (008), the S topologys the topology generated by the collecton of all sets of the form {t T : a / R (t G γ)} and {t T : a R (t G γ)} where G = (A g ) I, a A,andγ R. 0 The S topology on T s metrzable by the dstance d S, defned as follows. For each game G = (A g ) I,actona A,andtypet T,let h (t a G)= nf{γ : a R (t G γ)} 9 Ths means that σ (θ s ) = σ (θ t ) for all θ and all types s t wth the same (k )-order belefs. 0 The strategc topology can be gven an equvalent defnton that makes no drect reference to γ- ratonalzablty for γ 0. Indeed, by Ely and Pęsk (008, Lemma 4), a subbass of the strategc topology s the collecton of all sets of the form {t : a / R(t G 0)} and {t : a R (t G 0)}. Dekel et al. (006) defne the S topology drectly usng the dstance d S, rather than usng the topologcal defnton above.
454 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) Then, for each s and t T, d S (s t ) = m m sup G=(A g ) I G m max a A h (s a G) h (t a G) In terms of convergence of sequences, Dekel et al. (006) show that for every t T and every sequence (t n ) n n T,wehaved S (t n t ) 0 f, and only f, for every game G = (A g ) I,actona A,andγ R, the followng upper hemcontnuty (u.h.c.) and lower hemcontnuty (l.h.c.) propertes hold: For every sequence γ n γ, and for some sequence γ n γ, a R (t n G γ n ) n a R (t G γ) (u.h.c.) a R (t G γ) a R (t n G γ n ) n (l.h.c.) Dekel et al. (006) also ntroduce the unform-strategc (US) topology, whch strengthens the defnton of the strategc topology by requrng the convergence to be unform over all games. More precsely, the US topology s the topology of convergence under the metrc d US, whch s defned as d US (t s ) = sup G=(A g ) I G max h (t a G) h (s a G) a A Ths unformty renders the US topology partcularly relevant for envronments where the game both payoffs and acton sets s not fxed a pror, such as n a mechansm desgn envronment. We now ntroduce a metrc topology on types, whch we call unform-weak (UW) topology, under whch two types of player are close f they have smlar frst-order belefs, attach smlar probabltes to other players havng smlar frst-order belefs, and so on, where the degree of smlarty s unform over the levels of the belef herarchy. Thus, unlke the S and US topologes, whch are behavor-based, the UW topology s a belef-based topology,.e., a metrc topology defned explctly n terms of proxmty of herarches of belefs. The two man results of ths secton, Theorems and below, establsh a connecton between these behavor- and belef-based topologes. Before we present the formal defnton of the UW topology, recall that for a complete separable metrc space (S d), the topology of weak convergence on (S) s metrzable by the Prohorov dstance ρ, defnedas ρ(μ μ ) = nf{δ>0:μ(e) μ (E δ ) + δ for each measurable E S} μ μ (S) where E δ ={s S :nf s S d(s s )<δ}. generated by the dstance The UW topology s the metrc topology on T d UW (s t ) = sup d k (s t ) k s t T where d 0 s the dscrete metrc on and recursvely for k, d k s the Prohorov dstance on ( T k ) nduced by the metrc max{d 0 d k } on T k.
Theoretcal Economcs 5 (00) Unform topologes on types 455 In the remander of Secton 3 we explore the relatonshp between the UW topology and the S and US topologes. Frst, we show that the UW topology s fner than the US topology (Theorem ). Second, we prove a partal converse, namely that around fnte types,.e., types belongng to a fnte type space, the S topology (and hence also the US topology) s fner than the UW topology (Theorem ). 3. UW convergence mples US convergence Theorem. For each player I and for all types s t T, d US (s t ) 4Md UW (s t ) Thus the UW topology s fner than the US topology. Ths theorem s a drect mplcaton of the followng proposton. Proposton. Fx a game G, γ 0 and δ>0. Foreachntegerk, d k (s t )<δ R k (t G γ) R k (s G γ+ 4Mδ) I s t T The man challenge n provng ths result s due to the fact that (k )-order ratonalzable conjectures σ : T (A ) need not be contnuous under the topology of weak convergence of (k )-order belefs. Ths mples that, keepng the conjecture fxed, the ncentve constrants of player for k-order γ-ratonalzablty (cf. ()) may be dscontnuous n hs type under the topology of weak convergence of k-order belefs. Our proof overcomes ths ssue by endowng close-by types wth smlar, but not dentcal, conjectures. Indeed, the characterzaton of ICR from Proposton mples that for a gven acton a A and a gven mxed devaton α (A ),therealwaysexsts a (k )-order ratonalzable conjecture that s optmal to γ-ratonalze a aganst α at order k. Followng ths observaton, n our proof we endow type t wth an optmal conjecture for γ-ratonalzablty and endow type s wth an optmal conjecture for (γ + 4Mδ)-ratonalzablty. Usng these optmal conjectures, we then prove, usng an ntegraton-by-parts type argument, that every acton that s k-order γ-ratonalzable for t remans k-order (γ + 4Mδ)-ratonalzable for s. Proof of Proposton. Fx a game G = (A g ) I, γ 0 and δ>0. The proof s by nducton on k. For k =, lets and t T be such that d (s t )<δ. Fx an arbtrary a R (t G γ) and let us show that a R (s G γ + 4Mδ) usng Proposton. Fx α (A \{a }) and let σ : (A ) beaconjecturesuchthat 3 ( g (a σ (θ) θ) g (α σ (θ) θ) ) t θ] γ (4) θ To be precse, when we say that σ s an optmal conjecture to γ-ratonalze a aganst α at order k, we mean that σ s a (k )-order γ-ratonalzable conjecture that satsfes the followng property: for any type t, the expected payoff dfference between a and α for type t s at least γ under some (k )-order γ-ratonalzable conjecture f and only f ths expected payoff dfference s at least γ under σ. 3 Recall that t desgnates the frst-order belef of type t.
456 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) (Note that condton (3) s trval for k =.) Pck any functon a : A such that a (θ) arg max a A g (a a θ) g (α a θ)] θ and defne so that h(θ) = g (a a (θ) θ) g (α a (θ) θ) θ h(θ) g (a σ (θ) θ) g (α σ (θ) θ) θ (5) To conclude the proof for k =, we now show that θ h(θ)s θ] γ 4Mδ. Indeed, let {θ n } N n= be an enumeraton of such that h(θ n) h(θ n+ ) for all n N. Thus, t follows from d (s t )<δand h(θ) M for all θ that hence N h(θ)(s θ] t θ]) = (h(θ n ) h(θ n+ )) θ n= N = n= N δ n (s θ m] t θ n]) m= ( (h(θ n ) h(θ n+ )) s } {{ } {θ m } n m= ] t {θ m} n m= } {{ ]) } 0 δ n= h(θ n ) h(θ n+ ) = δ(h(θ ) h(θ N )) 4Mδ θ h(θ)s θ]= θ h(θ)(s θ] t θ]) + θ h(θ)t θ] 4Mδ+ θ h(θ)t θ] 4Mδ+ θ ( g (a σ (θ) θ) g (α σ (θ) θ) ) t θ] γ 4Mδ where the penultmate nequalty follows from (5) and the last nequalty follows from (4). Thus, a R (s G γ + 4Mδ) by Proposton, whch proves the desred result for k =. Proceedng by nducton, we now suppose the result s vald for some k and show that t remans vald for k +. Lets t T be such that d k+ (s t )<δ. Fx an arbtrary a R k+ (t G γ) and let us show that a R k+ (s G γ + 4Mδ). Fx α (A \{a }) and let σ : T k (A ) be a k-order γ-ratonalzable conjecture such that 4 T k ( g (a σ (θ t k ) θ) g (α σ (θ t k ) θ)) t k+ (dθ dt k ) γ (6) 4 Recall that t k desgnates the k-order belef of type t.
Theoretcal Economcs 5 (00) Unform topologes on types 457 Pck any measurable functon a : T k A such that a (θ t k ) arg max (g (a a θ) g (α a θ)) a R k (tk G γ+4mδ) (θ t k ) T k By constructon, a s a k-order (γ + 4Mδ)-ratonalzable conjecture. Thus, by Proposton,toconcludethata R k+ (s G γ+ 4Mδ), we need show only that ( g (a a (θ t k ) θ) g (α a (θ t k ) θ)) s k+ (dθ dt k ) γ 4Mδ (7) T k Let Ā ĀL be an enumeraton of the nonempty subsets of A and defne h l (θ) = max g (a a θ) g (α a θ)] θ l L a Āl Next, defne a partton {P P L } of T k as P l ={t k T k : Rk (tk G γ)= Āl} l L Snce σ s a k-order γ-ratonalzable conjecture, we have h l (θ) g (a σ (θ t k ) θ) g (α σ (θ t k ) θ) (θ tk ) P l and, therefore, L θ l= h l (θ)t k+ θ P l ] g (a σ (θ t k T k ) θ) g (α σ (θ t k ) θ)] t k+ (dθ dt k ) (8) Lkewse, defne a partton {Q Q L } as Q l ={t k T k : Rk (tk G γ+ 4Mδ) = Āl} l L Thus we have g (a a (θ t k ) θ) g (α a (θ t k ) θ)] s k+ (dθ dt k ) T k = θ whch, together wth (6)and(8), mples g (a a (θ t k ) θ) g (α a (θ t k ) θ)] s k+ (dθ dt k ) T k T k L l= h l (θ)s k+ θ Q l ] g (a σ (θ t k ) θ) g (α σ (θ t k ) θ)] t k+ (dθ dt k )
458 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) + θ L l= γ + θ h l (θ)(s k+ θ Q l ] t k+ θ P l ]) L h l (θ)(s k+ θ Q l ] t k+ θ P l ]) l= Therefore, to prove (7) and conclude that a R k+ (s G γ+ 4Mδ), we need only show that L h l (θ)(s k+ θ Q l ] t k+ θ P l ]) 4Mδ θ l= To prove ths nequalty frst note that the nducton hypothess mples Pl δ Q n l L (9) n:ān Āl Next, let N = L and consder an enumeraton {(θ n l n )} N n= of { L} such that for all n, and for all m, n, Thus, for each n = N, n ] θ m Q lm s k+ and, therefore, θ l= = L N n= m= h ln (θ n ) h ln+ (θ n+ ) (θ m = θ n and Āl m Āl n ) m n 5 (0) s k+ = s k+ t k+ n ] θ m Pl δ m m= ( n ) δ ] θ m P lm m= n ] θ m P lm m= h l (θ)(s k+ θ Q l ] t k+ θ P l ]) h ln (θ n )(s k+ θ n Q ln ] t k+ θ n P ln ]) (by (9) and(0)) δ (by d k+ (s t )<δ) 5 To see why an enumeraton of { L} that satsfes these two propertes exsts, note that t follows drectly from the defnton of h l (θ) that Āl Ām mples h l (θ) h m (θ).
Theoretcal Economcs 5 (00) Unform topologes on types 459 N = n= N = n= N δ (h ln (θ n ) h ln+ (θ n+ )) (h ln (θ n ) h ln+ (θ n+ )) } {{ } 0 n= n m= ( s k+ (s k+ θ m Q lm ] t k+ θ m P lm ]) n ] θ m Q lm m= t k+ n ]) θ m P lm m= } {{ } δ (h ln (θ n ) h ln+ (θ n+ )) = δh l (θ ) h ln (θ N )] 4Mδ as requred. Corollary. The Borel σ-algebras of the UW, US, S, and product topologes concde. Proof. Theorem mples that the Borel σ-algebra of the US topology s contaned n the Borel σ-algebra of the UW topology. Moreover, Lemma 4 n Dekel et al. (006) mples that the Borel σ-algebra of the strategc topology contans the product σ-algebra. Hence, t suffces to show that the product σ-algebra contans the UW σ-algebra. In effect, every unform-weak ball s a countable ntersecton of cylnders, therefore, every unform-weak ball s product-measurable, whch mples that every UW-measurable set s product measurable. An mportant mplcaton of ths corollary s that the Mertens Zamr unversal type space (T μ ) I remans a unversal type space when equpped wth any of the topologes S, US, or UW nstead of the product topology, a fact that was not known pror to ths paper. Indeed these topologes leave the measurable structure unchanged, so μ : T ( T ) remans the unque belef-preservng mappng and a Borel somorphsm, albet no longer a homeomorphsm. 3. S convergence to fnte types mples UW convergence Here we provde a partal converse to Theorem. We show that, as far as convergence to fnte types s concerned, convergence n the S topology mples convergence n the UW topology (and hence also n the US topology). Theorem. Around fnte types the S topology s fner than the UW topology,.e., for each player I, fnte type t T and δ>0 there exsts ε>0 such that for each s T, d S (s t ) ε d UW (s t ) δ Ths theorem s a drect mplcaton of Proposton 3 below, whch n turn reles on the followng result.
460 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) Lemma. Let (T ) I be a fnte type space. For every δ>0, there exst ε>0 and a game G = (A g ) I wth A T for all I, such that for every I and t T, t arg max g (a t θ)t θ t ] () a A t T θ and for every ψ ( A ) such that ψd] t D] δ for some D T, mn a A θ a A (g (t a θ) g (a a θ))ψθ a ] < ε () The proof of ths lemma, gven n the Appendx, uses a report-your-belefs game embedded n a coordnaton game. More precsely, we construct a game where each player chooses a pont n a fnte grd A ( T ) that ncludes all types n T (vewed as probablty dstrbutons over T ). If player chooses an acton n T the payoff to player s gven by a proper scorng rule, 6 7 whch guarantees that coordnatng on truthful reportng has the best-reply property, as shown n (). If, nstead, player chooses an acton n A \ T, then the payoff to player s no greater than the mnmum payoff under the scorng rule and strctly less when choosng an acton n T.Thus,fthegrdA ( T ) s suffcently fne, no acton t T can be an ε-best reply to a conjecture ψ ( A ) that s far from t (vewed as a probablty dstrbuton over A ), as shown n (). Indeed, ether ψ assgns large probablty to choosng an acton n A \ T, whch makes any a A \ T a proftable devaton, or t assgns enough probablty to T so that the condtonal ψ = ψ( T ) s close to ψ and hence far from t. Thus, n both cases, any grd pont a A \ T suffcently close to ψ s a proftable devaton. Proposton 3. Let (T ) I be a fnte type space. For each δ>0, there exst ε>0 and a game G such that for each nteger k, each player I,andeach(t s ) T T, d k (s t )>δ R (t G 0) R k (s G ε) 6 A proper scorng rule onameasurablespace s a measurable functon f : ( ) R such that f(ω μ)μ(dω) f(ω μ )μ(dω) for all μ, μ ( ), wth strct nequalty whenever μ μ. In the proof of the lemma, we use the scorng rule f : T ( T ) ] such that (θ t ψ) ψθ t ] ψ. 7 Dekel et al. (006) use a report-your-belefs game to prove ther Lemma 4, whch states that for every k and δ>0, thereexstsε>0 such that for all t s T, d k(s t ) δ mples d S(s t ) ε. However, t can be shown that, as k, the number of actons n ther game grows wthout bound and ε shrnks to 0. Thus, we cannot use a smlar constructon to prove our result. The game we construct dffers from thers n two respects: Frst, n our game the players report nfnte herarches of belefs, albet n a fnte type space, whereas n ther game players report only fntely many orders; second, Dekel et al. (006) use a pure report-your-belefs game, whle we embed a report-your-belefs game n a coordnaton game. The coordnaton feature ensures that the ratonalzable outcomes of our game hnge on nfntely many levels of the herarchy. Ths s mportant because when types fal to be close under d UW, there s no upper bound on the lowest order at whch the falure of proxmty occurs.
Theoretcal Economcs 5 (00) Unform topologes on types 46 Proof. Fx a fnte type space (T ) I and δ>0. Choose 0 <η<δsuch that for all k, I and t u T, 8 t k u k d k (t u )>η (3) By Lemma,thereexstε>0 and a game G = (A g ) I wth A T such that ()and () hold for every t T and every ψ ( A ) such that ψd] t D] η for some D T. Thus, for each (t s ) T T and each measurable functon σ : T (A ),fforsomed T, σ (θ s )a ]s (θ ds ) t D] η (θ a ) D T } {{ } ψ(θ a ) then for some a A, T g (t σ (θ s ) θ) g (a σ (θ s ) θ) ] s (dθ ds )< ε (4) We now show that for each I, t R (t G 0) t T (5) d k (s t ) η t / R k (s G ε) k (t s ) T T (6) For I and t T consder the conjecture σ : T (A ) wth σ (θ t )t ]= for all (θ t ) T.Thenactont s a best reply for type t to conjecture σ by (), hence t R (t G 0) by the characterzaton of ICR n terms of best-reply sets, thus provng (5). To prove (6) fork =, pcks T wth d (s t ) η. Then there exsts E such that s E] t E] η and, hence, for every σ : T (A ), lettng D = E T, σ (θ s )a ]s (θ ds ) = σ (θ s )T ]s (θ ds ) (θ a ) D T θ E T s E] t E] η = t D] η It follows from (4) thatt / R (s G ε). Proceedng by nducton, let k and assume that (6) holdsfor k. Fx I and t T,andpcks T wth d k(t s ) η. Then there exsts some E π k (T ) wth s k Eη ] t k E] η (7) Defne D ={(θ t ) T : (θ t k ) E}, sothatt D]=t k E]. Consderanarbtrary (k )-order ε-ratonalzable conjecture σ : T (A ),.e., supp σ (θ s ) R k (s G ε) (θ s ) T 8 Such postve η exsts because, gven any fnte type space (T ) I, there exsts K such that d k (t u ) = d K (t u ) for all k K and t u T.
46 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) By the nducton hypothess and the condton above, Thus, d k (s t ) η σ (θ s )t ]=0 (θ s t ) T T (8) (θ a ) D T σ (θ s )a ]s (θ ds ) = (θ t k ) E (θ t k ) E (θ t k ) E s k σ (θ s )T (π k T (π k ) ({t k } η ) ) (t k )]s (θ ds ) σ (θ s )T (π k ) (t k )]s (θ ds ) θ {tk } η ]=s k Eη ] t k E] η = t D] η where the frst nequalty follows from (8), the second equalty follows from (3), and the last nequalty follows from (7). By (4), ths mples t / R k (s G ε). Theorems and combned yeld the followng corollary. Corollary. The UW, US, and S topologes are all equvalent around fnte types. To end ths secton, we remark that n Theorem we cannot dspense wth the assumpton that t s a fnte type. Indeed, n the next secton we prove that the US topology s strctly fner than the S topology. Thus, the UW topology cannot be equvalent to the S topology, for we have shown that the UW topology s fner than the US topology (Theorem ). A more drect way to argue that the UW topology s strctly fner than the S topology s to note that the unversal type space s not separable under the UW topology (a result that s nterestng n ts own rght), whereas Dekel et al. (006) show that a countable set of fnte types s dense under the strategc topology. To see why the unform-weak topology s not separable, fx two states θ 0 and θ n, and consder the nonredundant type space (X ) I,whereX ={0 } N and each type x = (x n ) n N assgns probablty to the par (θ x L (x )), wherel : X X s the shft operator,.e., L((x x )) = (x x 3 ) for each x = (x n ) n N. Clearly, the UW dstance between any two dfferent types n X s and, hence, under the UW metrc, X s a dscrete subset of the unversal type space. Snce X s uncountable, t follows that the unversal type space s not separable under the UW topology. 4. Nongenercty of fnte types Dekel et al. (006) show that fnte types are dense under the S topology, thus strengthenng an early result of Mertens and Zamr (985) that fnte types are dense under the
Theoretcal Economcs 5 (00) Unform topologes on types 463 product topology. In contrast, n Theorem 3 below we show that under the US topology, fnte types are nowhere dense,.e., the closure of fnte types has an empty nteror. 9 An mplcaton of ths result and Theorem s that the US topology s strctly fner than the S topology. 30 The proof of Theorem 3 reles on Lemmas and 3 below. Lemma states that fnte types are not dense under the UW topology. To prove ths, we consder an nstance of the countably nfnte common-pror type space from Rubnsten s (989) E-Mal game and show that none of ts types can be UW-approxmated by a sequence of fnte types. In Lemma 3 we show that any sequence of types that fals to converge to a type n the E-Mal type space under the UW topology must also fal to converge under the US topology. Together, these lemmas mply that fnte types are bounded away from the E-mal type space n US dstance, whch we state as Proposton 4 below. Ths mples that the set of fnte types s not dense under the US topology. Usng ths result, the proof of Theorem 3 shows that every fnte type can be US-approxmated by a sequence of nfnte types, none of whch s the US lmt of a sequence of fnte types, thereby establshng nowhere denseness. In effect, consder the followng nstance of the E-Mal type space. Let ={θ 0 θ } and let the type space (U U ) be 3 U ={u 0 u u } U ={u 0 u u } where u 0 θ 0 u 0 ]=, u 0 θ 0 u 0 ]=/3, u 0 θ u ]=/3, u n θ u n ]=/3 u n θ u n ]=/3 n u n θ u n ]=/3 u n θ u n+ ]=/3 n We have the followng result. Proposton 4. For every I, fnte type t T,andn 0, d US (t u n ) M/6. The proposton s a drect consequence of the followng two lemmas. Lemma. For every I, fnte type t T,andn 0, d UW (t u n ) /3. Lemma 3. For every I, t T,andn 0, d US (t u n ) (M/)d UW (t u n ). 9 Ths s equvalent to sayng that the complement of the set of fnte types contans an open and dense set under the US topology. 30 Dekel et al. (006) state the result that the US topology s strctly fner than the S topology. However, as reported n Chen and Xong (008), the proof n that paper contans a mstake. 3 Ths type space s an nstance of the E-Mal type space where the more nformed player who receved k messages attaches probablty p = /3 (resp. p = /3) toplayer havng receved k (resp. k) messages, and the less nformed player who receved k messages attaches probablty p (resp. p)toplayer havng receved k (resp. k + ) messages. Our choce that p = /3 s mmateral; our results hold true f we assume any other value for p.
464 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) Fgure. ThegamefromLemma 3 for N = and M = 4. In the proof of Lemma, gven n the Appendx, we frst show that the UW dstance between any two dstnct types of any player n the E-Mal type space above s at least /3. 3 Second, we show that any fnte type t n whose UW dstance from u n s less than /3 must attach postve probablty to (and hence mples the exstence, n the same fnte type space, of) a type t n+ whose UW dstance from u n+ s less than /3,whch n turn mples the exstence n the same fnte type space of some type t n+ whose UW dstance from u n+ s less than /3 and so on. These two facts together mply the contradcton that the types t t are all dfferent but belong to the same fnte type space, whence the result follows. Turnng to Lemma 3, the proof, also n the Appendx, constructs, for each δ 0 and N 0, a game such that for each 0 n N, a certan acton a n s ratonalzable for u n but s not δ-ratonalzable for any type t wth d k (t u n )>δ/m, where the order k grows wth the dfference N n. To provde ntuton, we sketch the argument for the case N =. The game correspondng to ths case s depcted n Fgure, wththepayoff bound normalzed to M = 4. It s clear that n ths game, for all = and n = 0, actona n s ratonalzable for u n. 33 However, a n s weakly domnated by s, and the payoffs from b n and c n are such that whenever the belefs of a type t are suffcently far from those of u n,thenany δ-ratonalzable conjecture about player that δ-ratonalzes a n aganst s cannot do so aganst both b n and c n as well. Indeed, we have d k (t u n )>δ/m a n / R k (t δ) k n (9) To see ths for k =, frst note that a 0 s weakly domnated by s,hencea 0 / R (t δ) for any type t wth d (t u 0 )>δ/. Indeed, u 0 θ 0]= and hence d (t u 0 )>δ/ mples t θ 0] < δ/, so the hghest possble expected payoff for t under a 0 s δ, whereas s yelds 0. By the same token, a / R (t δ) for any t wth d (t u )>δ/ and a / R (t δ) for any t wth d (t u )>δ/. Consderactona 0 now and pck any t such that d (t u 0 )>δ/. Snce u 0 θ ]=/3, wemusthaveethert θ ] < 3 The type u k of player who receved k messages assgns probablty /3 to the other player havng receved k messages, whle u k+ attaches probablty 0 to that event, and smlarly for player. 33 The par (ς ς ) wth ς (u n ) = a n f n and ς (u n ) = s f n has the best-reply property.
Theoretcal Economcs 5 (00) Unform topologes on types 465 /3 δ/ or t θ 0] < /3 + δ/. Pck any conjecture σ that δ-ratonalzes a 0,sothatthe dfference n expected payoff between s and a 0 s at most δ.thsrequresthenduced dstrbuton over A to satsfy Prθ 0 a 0 t σ ]+Prθ a t σ ] δ/4 hence the dfference n expected payoffs between b 0 and a 0 s Prθ 0 a 0 t σ ] Prθ a t σ ] 3 Prθ a t σ ]+ δ/4 whch s greater than δ when t θ ] < /3 δ/. Lkewse, the dfference n expected payoffs between c 0 and a 0 s Prθ 0 a 0 t σ ]+ Prθ a t σ ] 3 Prθ 0 a 0 t σ ]+ δ/ whch s greater than δ when t θ 0] < /3+δ/. Thus, n any case, a 0 / R (t δ)and the proof of (9)fork = s complete. The proof for n = 0 and k = uses the arguments just gven for the case k = and s completely analogous for nstance, those arguments show that f σ s a frst-order δ-ratonalzable conjecture that δ-ratonalzes a 0 for a type t,thenwemusthave δ/4 Prθ 0 a 0 t σ ] t θ 0 {u 0 }δ/m ] and hence the dstance between the second-order belefs of t and u 0 s at most δ. We are now ready to prove the man result of ths secton. Theorem 3. Fnte types are nowhere dense under the US and the UW topology. Proof. It suffces to prove that every fnte type can be UW-approxmated by a sequence of nfnte types, none of whch s the US lmt of a sequence of fnte types. 34 Fxafntetypespace(T T ) and a type t T.Foreachn, letδ n = /(n + ) and defne the nfnte type t n by the requrement that, for every k and every measurable E T k, t k n E]=( δ n)t k E]+δ nu k 0 E] Note that for all n, k, and measurable E T k,wehave t k n E]=( δ n)t k E]+δ nu k 0 E] tk n Eδ n ]+δ n hence d UW (t n t ) δ n 0. It remans to prove that none of the types n the sequence (t n ) n s n the US closure of the set of fnte types,.e., for every n, thereexstsε n > 0 such that the US dstance between t n and every fnte type n T s at least ε n.thus,fxn, pck any 0 <ε n < mn{m/6 M/(3n + )}, any fnte type space (S S ),andanytypes S,and let us show that d US (t n s ) ε n. Usng Lemma, choose N large enough so that d (N+) (t u 0 ) /3 t T S (0) 34 Indeed, by Theorem, the sequence also US-approxmates the fnte type, hence nowhere denseness n the US topology follows. By the same theorem, none of the types n the sequence wll be the UW lmt of a sequence of fnte types, thus nowhere denseness n the UW topology also follows.
466 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) and let G N = (A N g N ) = be the game defned n the proof of Lemma 3. Nowdefne another game G N = (A N g N ) = as A N = A N A N = A N {0 } and for all a A N, a A N, x {0 } and θ, g N (a a x θ)= g N(a a θ) M/ f x = and a = a 0 g N (a a x θ)= g N(a a θ)+ M/(3n + ) f x = and a a 0 0 otherwse. Note that snce all payoffs n G N are between M and M, the same s true for all payoffs n G N. Moreover, we have the followng lemma, whch s proved n the Appendx. Lemma 4. For all k 0 and all ε 0, R k (t G N ε) = R k (t G N ε) t T () R k (t G N ε) = proj A N R k (t G N ε) t T () We now prove that (a ) R (t n G N 0) for some a A N,but(a ) / R (s G N ε n) for all a A N, reachng the desred concluson that d US(t n s ) ε n. To show that (a ) R (t n G N 0) for some a A N, t suffces to construct a ratonalzable conjecture σ n game G N under whch, for all a A N, actons (a 0) and (a ) gve t n the same expected payoff. Let σ : T (A N ) be an arbtrary ratonalzable conjecture n G N and defne σ : T (A N ) as σ (θ t )a ]=σ (θ t )a ] t T \ U a A N σ (θ u k)a k ]= k 0 From the proof of Lemma 3, tfollows,usng() wthε = 0, thatσ s a ratonalzable conjecture n G N and also, usng (0)andthefactthatε n <M/6, t follows that a 0 / R (t G N ε n ) t T S (3) Thus, σ (θ t )a 0 ]=0 for all θ and t T,henceforalla A N we have T g N (σ (θ t ) a θ) g N (σ (θ t ) a 0 θ) ] t n (dθ dt ) = δ n 3 ( M δ ) n M 3 3n + = 0 Ths proves that (a 0) and (a ) gve type t n the same expected payoff under σ for all a A N,aswastobeshown.
Theoretcal Economcs 5 (00) Unform topologes on types 467 Turnng to the proof that (a ) / R (s G N ε n) for all a A N,consderanarbtrary ε n -ratonalzable conjecture σ n game G N. By () and(3), for all θ and s S,wemusthaveσ (θ s )a 0 ]=0. Thus, for all a A N, s θ s ] g N (σ (θ s ) a θ) g N (σ (θ s ) a 0 θ) ] = M 3n + < ε n (θ s ) S whch proves that (a ) s not ε n -ratonalzable for s n game G N. 5. Dscusson 5. Relaton wth common p-belefs As we mentoned n the Introducton, the unform-weak topology s related to the noton of common p-belef due to Monderer and Samet (989). Fx a state θ and p 0 ]. For each player I, defne B p (θ) ={t T : t θ] p} and Bk p (θ) = { t k T k : t k θ Bk p (θ)] p } recursvely for all k. Atypet has common p-belef n θ, and we wrte t C p (θ), f t k B k p (θ) for all k. A sequence of types (t n ) n has asymptotc common certanty of θ f for every p<,wehavet n C p (θ) for n large enough. Monderer and Samet (989) use ths noton of proxmty to common certanty,.e., common -belef, to study the robustness of Nash equlbrum to small amounts of ncomplete nformaton. Ther man result states that for any game and any sequence of common-pror type spaces, a suffcent condton for Nash equlbrum to be robust to ncomplete nformaton (relatve to the gven sequence of type spaces) s that for some sequence p n, the pror probablty of the event that the players have common p n - belef on the payoffs from the complete nformaton game converges to as n.arelated paper, Kaj and Morrs (997), shows that asymptotc common certanty s actually a necessary condton for robustness n all games. Snce both results are formulated for Bayesan Nash equlbrum n common-pror type spaces, to facltate comparson wth our results, we report (wthout proof) an analogue of ther results for nterm correlated ratonalzablty wthout mposng common prors. Proposton 5. A sequence of types (t n ) n has asymptotc common certanty of θ f and only f for every game and every ε>0, every acton that s ratonalzable for player when θ s common certanty remans nterm correlated ε-ratonalzable for type t n for all n large enough. Thus the only f part s an nterm verson of Monderer and Samet (989, Theorem B ) and the f part s an nterm verson of Kaj and Morrs (997, Proposton 0). As t turns out, the unform-weak topology can be vewed as an extenson of the concept of asymptotc common certanty: these two notons of convergence concde when the lmt type has common certanty of some state. Indeed, lettng t θ desgnate the type of player who has common certanty of θ, we can make the followng proposal.
468 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) Proposton 6. A sequence (t n ) n 0 has asymptotc common certanty of θ f and only f (t n t θ ) 0 as n. d UW Proof. It suffces to show that for each I, p 0 ], andk, wehaveb k p (θ) = {t θ k } p.fork =, ths follows drectly from t θ θ]=. Now suppose ths holds for k and let us show that t also holds for k. Indeed, B k p (θ) = { t k T k : t k θ Bk p (θ)] p } = { t k T k : t k θ {tk θ } p ] p } ={t θ k } p where the second equalty follows from the nducton hypothess and the thrd equalty follows from the fact that t θ k θ tk θ ]=. Thus, taken together, Theorems and extend Proposton 5 to envronments where the lmt type has nondegenerate ncomplete nformaton. 35 5. Other unform metrcs The Prohorov metrc, on whch the unform-weak topology s based, s but one of many equvalent dstances that metrze the topology of weak convergence of probablty measures. For any such dstance, one can consder the assocated unform dstance over herarches of belefs. Interestngly, these metrcs can generate dfferent topologes over nfnte herarches, even though the nduced topologes over k-order belefs concde for each k. Below we provde such an example. Gven a metrc space (S d), letbl(s d) desgnate the vector space of real-valued, bounded, Lpschtz contnuous functons over S, endowed wth the norm { f BL = max sup f(x) sup x x y f(x) f(y) d(x y) } f BL(S d) Recall that the bounded Lpschtz dstance over (S d) s { } β(μ μ ) = sup fdμ fdμ : f BL(S d) wth f BL μ μ (S d) Ths dstance metrzes the topology of weak convergence and t relates to the Prohorov metrc ρ as 36 (/3)ρ β ρ Now defne a unform metrc β UW over herarches of belefs as follows. Let β 0 denote the dscrete metrc over and, recursvely, for k, letβ k denote the bounded Lpschtz metrc on ( T k ) when T k s equpped wth the metrc max{β 0 β k }. Then 35 Note that t θ s a fnte type. 36 See Dudley (00, pp. 398 and 4). β UW = sup β k k
Theoretcal Economcs 5 (00) Unform topologes on types 469 For each k,themetrcβ k s equvalent to dk, as they both nduce the weak topology on k-order belefs. However, as we now show, β UW s not equvalent to d UW. 37 Suppose that ={θ 0 θ } and for each n, consder the type space (T n ) I,where T n ={u 0 u t n } I and belefs are u 0 θ 0 u 0 ]= u θ u ]= I and t n θ 0 u 0 ]=/n t n θ t n ]= /n I Thus d k(t n u ) = /n for all k and, therefore, d UW (t n u ) 0 as n. We nowshowthatβ UW (t n u ) 0. Letf be the ndcator functon of {θ },.e., f(θ m ) = m for m {0 }. Thendefnethek-order terated expectaton of f for each k and each player, denoted f k : T k R,as f (t ) = fdt = t θ ] and f k (tk ) = f k dt k for k Thus, we have f k du k = and f k dt n k = ( /n)k Snce t can be shown that f k BL(T k β k ) and f k BL, wehaveβ k (t n u ) ( /n) k and hence β UW (t n u ) for every n. Ths example s also relevant for the comparson between our work and Morrs (00), who shows that the topology of unform convergence of terated expectatons s equvalent to the strategc topology assocated wth a restrcted class of games, called hgher-order expectatons (HOE) games. By ths result and the example above, unformweak convergence s not suffcent for convergence n the strategc topology for HOE games. Ths mght seem puzzlng at frst, gven that unform-weak convergence has been shown to mply convergence n Dekel et al. (006) strategc topology, whch s defned by requrng lower hemcontnuty of the strct ICR correspondence n all games, not just HOE games. To reconcle these facts, we note that the noton of strct ICR correspondence mplctly used n Morrs (00) s dfferent from the one we use, n that t does not requre the slack n the ncentve constrants to hold unformly n a bestreply set. Thus, for a gven game, contnuty of Morrs (00) noton of strct ICR s more demandng than ours. 37 The example below actually shows that the two metrcs are not equvalent even around complete nformaton types. In partcular, asymptotc common certanty does not guarantee convergence under β UW.
470 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) 6. Concluson Our results shed lght on the connecton between smlarty of belefs and smlarty of behavors n games, but leave open a number of nterestng questons for future research. One queston s whether unform-weak convergence s also a necessary condton for unform-strategc convergence. We beleve the answer s n the affrmatve and are pursung ths conjecture n ongong research. Ths queston s of partcular nterest because of the tenson between Theorem and Theorem 3, whch mply that the unform-weak and the unform-strategc topologes are equvalent around types n a nowhere dense set. Another mportant avenue of research s to characterze the (nonunform) strategc topology n terms of proxmty of belefs; we are also explorng ths queston n ongong work. Fnally, t would be nterestng to examne strategc topologes for soluton concepts that refne ICR, such as Bayesan equlbrum, ncomplete nformaton versons of correlated equlbrum, or nterm ndependent ratonalzablty. Appendx: Omtted Proofs Proof of Proposton. Fxk, t T,anda A.Let denote the set of equvalence classes of measurable functons σ : T (A ) such that supp σ (θ t ) R k (t G γ) for t -almost every (θ t ) T, where we dentfy pars of functons that are equal t -almost surely. The set can be vewed as a compact convex subset of the topologcal vector space L of (equvalence classes of) R A -valued measurable functons over T. 38 Consder the functon F : (A \{a }) R such that F(α σ ) = g (a σ (θ t ) θ) g (α σ (θ t ) θ) ] t (dθ dt ) T Thus, F s the restrcton of a contnuous blnear functonal on R ( A ) L to the Cartesan product of compact, convex sets. By a mnmax theorem of Fan (953), mn α (A \{a }) max F(α σ ) = σ max σ mn F(α σ ) α (A \{a }) 38 The space L s equpped wth the weak* topology nduced by the probablty measure t ( T ). Under ths topology, a sequence (f n ) n N n L converges to f L f and only f for each contnuous and bounded functon h : T R A, h(θ t ) f n (θ t ) t (dθ dt ) h(θ t ) f (θ t ) t (dθ dt ) as n where desgnates the Eucldean nner product n R A. To see why s compact, note that the dsntegraton property of probablty measures yelds a natural homeomorphsm between and { ν ( T A ) : ν ( {(θ t a ) : a R k (t G γ)} ) } = marg T ν = t whch s a closed subset of the compact space ( T A ).
Theoretcal Economcs 5 (00) Unform topologes on types 47 Now, a R k (t G γ) f and only f the rght-hand sde s greater than or equal to γ. Thus, a R k (t G γ)f and only f for every α (A \{a }),thereexstsσ such that F(α σ ) γ, whch s the desred result. Proof of Lemma. For each I, letρ and denote the Prohorov dstance on ( T ) and the Eucldean norm on R T, respectvely. Also, let f : T ( T ) R be the functon defned by f (θ t ψ)= ψθ t ] ψ and let F : ( T ) ( T ) R be the functon defned by F (ψ ψ)= (θ t ) T f (θ t ψ )ψθ t ] Note that F (ψ ψ) F (ψ ψ)= ψ ψ for all ψ ψ ( T ),hence and also 39 η mn{ F (ψ ψ) F (ψ ψ): ψ ψ ( T ) ρ (ψ ψ ) δ} > 0 ρ (ψ ψ )<η/ F (ψ ψ) F (ψ ψ)<η ψ ψ ( T ) The compact set ( T ) can be covered by a fnte unon of open balls of radus η/. (These balls are taken accordng to the metrc ρ.) Choose one pont n each of these balls and let A ( T ) denote the fnte set of selected ponts. Enlarge A,f necessary, to ensure A T. (Recall that we dentfy each t T wth μ (t ).) Thus, for every ψ ( T ),thereexstsa A \ T such that F (ψ ψ) F (a ψ)<η. Now defne the payoff functon g : A A R,as f (θ a a ) f a T g (θ a a ) = 4/δ f a T and a / T f a / T and a / T. It follows drectly from the defnton of g and the fact that t T ]= that each a A yelds an expected payoff of F (a t ) to type t under the conjecture σ : T (A ) such that σ (θ t )t ]= for all (θ t ) T. Snce F (t t ) F (a t ) for all a A,() follows. Fx any 0 <ε<mn{η( δ/) δ/}. We shall prove () now. Fxt T and ψ ( A ), and assume that there exsts D T such that ψd] t D] δ. Frst 39 Lettng h : T ] denote the mappng (θ t ) h(θ t ) = ψθ t ] ψ θ t ],foreach ζ 0,wehave F (ψ ψ) F (ψ ψ)= ψ ψ = ψθ t ]h(θ t ) ψ θ t ]h(θ t ) ζ (θ t ) T (θ t ) T whenever ρ (ψ ψ ) ζ.
47 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) suppose ψ T ] < δ/. Pck any a A \ T. Snce f maps nto ], (g (t a θ) g (a a θ))ψθ a ] ( δ/) + (δ/)( 4/δ + ) a A θ = δ/ < ε whch proves () for the case ψ T ] < δ/. Now suppose that ψ T ] δ/. Consder the condtonal probablty ψ( ) ψ( T ).Then hence ψd] ψd]= ψd]ψ T ] ψd] δ/ ψd] t D] ψd] t D] ψd] ψd] δ δ/ = δ/ whch mples F ( ψ ψ) F (t ψ) η by the defnton of η. Nowpckanya A \ T wth ρ ( ψ a )<η/,sothatf (a ψ) F ( ψ ψ) > η. ThenF (a ψ) F (t ψ) > η and hence (g (t a θ) g (a a θ))ψθ a ] a A θ = (F (t ψ) F (a ψ))ψ T ]+( 4/δ + )( ψ T ]) (F (t ψ) F (a ψ))ψ T ] <( δ/)( η) < ε whch proves () also for the case ψ T ] δ/. 40 Proof of Lemma. Frst we prove by nducton that d UW (u n u m ) /3 = n 0 m 0 s.t. m n (4) For all n we have u 0 θ 0]= and u n θ 0]=0, henced (u 0 u n ) = > /3; moreover, u 0 θ 0]=/3 and u n θ 0]=0, henced (u 0 u n ) /3. Assume that we have proved d n(u n u m ) /3 for all =, somen, all n N, andallm n. Then, for all m>n, snce u n θ u n ]=/3 and u m θ u l ]=0 for all l<n,we obtan u n+ n θ u n n ]=/3 and un+ m θ {u n n }/3 ]=0, henced n+ (u n u m ) /3. Snce u n θ u n ]=/3 and u m θ u l ]=0 for all l n, we also get u n+ n θ u n n ]=/3 and un+ m θ {u n n }/3 ]=0,henced n+ (u n u m ) /3. The proof of (4)scomplete. Now let (T T ) beafntetypespace,andforevery = and every n 0,defne T n ={t T : d UW (t u n )</3} We must show that each T n s empty. Note that (4) mplest n T m = for each player,andalln 0 and m 0 such that m n. Thus, t suffces to show that f T n 40 To ensure that the payoffs are bounded by M, we can multply g and ε by a factor of Mδ/4, f necessary. Ths normalzaton does not affect the valdty of ().
Theoretcal Economcs 5 (00) Unform topologes on types 473 for some player and some n 0, thent m and T m for all m>n,asths contradcts the fnteness of T and T. Assume that T 0. Pck any t 0 T 0 and /3 >δ>d UW (t 0 u 0 ).Then t k 0 θ 0 {u k 0 }δ ] u k 0 θ 0 u k 0 ] δ = δ k and hence, usng the fact that δ</3 and t 0 θ 0 T ]=t 0 θ 0 T ],also t 0 θ 0 T 0 ] t 0 θ0 {t T : d UW (t u 0 )<δ} ] δ>0 mplyng that T 0 as well. Now let n 0 and assume T n. Pck any t n T n and /3 >δ>d UW (t n u n ).Then and hence, as before, t n k θ {u k n+ }δ ] u k n θ u k n+ ] δ = /3 δ k t n θ T n+ ] t n θ {t T : d UW (t u n+ )<δ} ] /3 δ>0 so T n+. Smlarly, we can show that T n mples T n for all n. Proof of Lemma 3. Forany gvenn we construct a game G N wth acton sets such that A N ={a 0 a b c a N b N c N s } A N ={a 0 b 0 c 0 a N b N c N a N s } and, moreover, for every δ 0 and 0 k N, a n R (u n G N 0) I 0 n N (5) a n R (k+) (t G N δ) d (k+) (t u n ) δ/m n N k t T (6) a n R k+ (t G N δ) d k+ (t u n ) δ/m n N k t T (7) Indeed, ths mples the statement of the lemma. Fx N. For convenence, throughout the proof let a N+ = s and θ n = θ for every n. The payoffs n G N are as follows. Actons s and s gve constant payoffs g N (θ s a ) = g N (θ a s ) = 0 for every θ, a A N,anda A N Actons a 0 a N and a 0 a N are weakly domnated by s and s, respectvely: 0 f n = 0 and (θ a ) = (θ 0 a 0 ) g N (θ a n a ) = 0 f n>0 and (θ a ) {(θ a n ) (θ a n )} M otherwse { 0 f (θ a ) {(θ g N (θ a a n ) = n a n ) (θ a n+ )} M otherwse.
474 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) The payoffs for actons b c b N c N are { M/4 f (θ a ) = (θ a n ) g N (θ b n a ) = g N (θ c n a ) = M/ f (θ a ) = (θ a n ) g N (θ b n a ) = g N (θ c n a ) = M otherwse. Fnally, the payoffs for b 0 c 0 b N c N are { M/4 f (θ a ) = (θ n a n ) g N (θ a b n ) = g N (θ a c n ) = M/ f (θ a ) = (θ a n+ ) g N (θ a b n ) = g N (θ a c n ) = M otherwse. It s mmedate to verfy that (5) holds. To see ths, just note that the mappngs ς : U A N such that ς (u n ) = a n for 0 n N and ς (u n ) = s for n>n have the best reply property. It remans to prove that (6)and(7) hold for every 0 k N. Todoths,wenowfx δ 0 and establsh the followng three clams. Frst, we show that (7) holds for k = 0. Second, we prove that (7) mples (6)forall0 k N. Thrd, we show that f (6) holds for some 0 k < N, then(7) holds wth k + substtuted for k, thus concludng the proof. To ease notaton, for every player, typet T, and conjecture σ : T (A N ), n what follows we wrte Pr t σ ] for the probablty dstrbuton over A N nduced by t and σ,.e., Prθ a t σ ]= σ (θ t )a ]t (θ dt ) (θ a ) A N T To prove our frst clam, namely that (7) s vald for k = 0, fxanyt T and 0 n N, assume that a n R (t G N δ),andletσ : T (A N ) be a correspondng 0-order δ-ratonalzable conjecture. Snce a n s a δ-best reply to σ, the dfference n expected payoff when choosng s nstead of a n under σ must be at most δ,hence Prθ n a n t σ ]+Prθ a n+ t σ ] δ/m (8) Smlarly, the dfference n expected payoff when choosng b n or c n nstead of a n under σ must be at most δ,hence δ 4 M Prθ n a n t σ ] M Prθ a n+ t σ ] δ The latter nequaltes together wth (8)mply Prθ n a n t σ ] /3 δ/m Prθ a n+ t σ ] /3 δ/m (9) hence t θ n] /3 δ/m and t θ ] /3 δ/m.moreover,fn>0,then(8)mples t θ ] δ/m.thus,d (t u n ) δ/m,as(7) requres for k = 0. To prove our second clam, namely that (7) mples (6) forall0 k N, fxany such k, anyt T,andany0 n N, assume that a n R (k+) (t G N δ),andlet σ : T (A N ) be a correspondng (k + )-order δ-ratonalzable conjecture.
Theoretcal Economcs 5 (00) Unform topologes on types 475 Frst consder the case n = 0. Snce a 0 s a δ-best reply to σ,tmustgveanexpected payoff wthn δ of the one from s,hence Prθ 0 a 0 t σ ] δ/m δ/m Snce σ s (k + )-order δ-ratonalzable, from (7) we thus obtan t (k+) θ 0 {u k+ 0 } δ/m ] δ/m as requred by (6) whenn = 0. Next consder the case n>0. Snce a n s a δ-best reply to σ, t must gve an expected payoff wthn δ of the one from s,hence Prθ a n t σ ]+Prθ a n t σ ] δ/m Smlarly, comparng a n to b n and c n,wemusthave δ 4 M Prθ a n t σ ] M Prθ a n t σ ] δ The latter three nequaltes together mply Prθ a n t σ ]+Prθ a n t σ ] δ/m (30) Prθ a n t σ ] /3 δ/m (3) Prθ a n t σ ] /3 δ/m (3) Snce σ s (k + )-order δ-ratonalzable, by (7) wehaveσ (θ t )a n ]=0 for all t T such that d k+ (t u n )>δ/m and σ (θ t )a n ]=0 for all t T such that d k+ (t u n )>δ/m.by(30), (3), and (3) ths mples t (k+) θ {u k+ n uk+ n }δ/m ] δ/m t (k+) θ {u k+ n }δ/m ] /3 δ/m t (k+) θ {u k+ }δ/m ] /3 δ/m n as requred by (6)whenn>0. It remans to prove our thrd clam. Assumng (6) forsome0 k<n,wemust show that (7) remans vald when k s replaced by k +. Pck any t T and 0 n N k, assume that a n R (k+)+ (t G N δ),andletσ : T (A N ) be a correspondng (k + )-order δ-ratonalzable conjecture. Snce a n s a δ-best reply to σ, the dfference n expected payoff when choosng s or b n or c n nstead of a n under σ must be at most δ. Thus, as before, (8)and(9) must hold. Moreover, snce σ s (k + )-order δ-ratonalzable, by (6) wehaveσ (θ n t )a n ]=0 for all t T wth d (k+) (t u n )>δ/m and σ (θ t )a n+ ]=0 for all t T wth d (k+) (t u n+ )> δ/m. Ths mples θ n {u (k+) n } δ/m ] /3 δ/m t (k+)+ t (k+)+ θ {u (k+) n+ }δ/m ] /3 δ/m
476 Chen, D Tllo, Fangold, and Xong Theoretcal Economcs 5 (00) and, f n>0,also t (k+)+ θ {u (k+) n u (k+) n+ }δ/m ] δ/m as requred by (7)whenk s replaced by k +. Proof of Lemma 4. Fx ε 0 and note that () and() are trvally true for k = 0. Now we assume they are true for some k 0 and prove that they hold for k +. Note that snce () holds for k, there exsts a mappng ξ : T A N {0 } that satsfes (a ξ(t a )) R k (t G N ε) t T a R k (t G N ε) (33) Let us prove () fork + now. Fx any t T and a R k+ (t G N ε), andlet σ : T (A N ) be a correspondng k-order ε-ratonalzable conjecture. Defne the conjecture σ : T (A N ) for game G N as σ (θ t )a ξ(t a )]=σ (θ t )a ] θ t T a A N By (33), σ s a k-order ε-ratonalzable conjecture. Moreover, the dfference n expected payoff for t between any a A N and a under σ n game G N s T k = g N (a σ (θ t ) θ) g N (a σ (θ t ) θ) ] t k+ (dθ dt k ) T k g N (a σ (θ t ) θ) g N (a σ (θ t ) θ) ] t k+ (dθ dt k ) ε = ε where the nequalty follows from the fact that a R k+ (t G N ε). Thsprovesthat a R k+ (t G N ε), and we have thus shown that Rk+ (t G N ε) R k+ (t G N ε). Conversely, pck any a R k+ (t G N ε)and let σ : T (A N ) be a correspondng k-order ε-ratonalzable conjecture. Defne σ : T (A N ) as σ (θ t ) = marg A N σ (θ t ) θ t T Snce () holds for k, thssak-order ε-ratonalzable conjecture n G N. Moreover, the dfference n expected payoff for t between any a A N and a under σ n game G N s T k g N (a σ (θ t ) θ) g N (a σ (θ t ) θ) ] t k+ (dθ dt k ) = g N (a T k σ (θ t ) θ) g N (a σ (θ t ) θ) ] t k+ (dθ dt k ) ε hence a R k+ (t G N ε). Ths shows that R k+ (t G N ε) Rk+ (t G N ε),sothe proof of ()fork + s complete.
Theoretcal Economcs 5 (00) Unform topologes on types 477 Nowweshowthat() also remans true for k +, thus concludng the proof. Fx t T,leta R k+ (t G N ε), andletσ : T (A N ) be a correspondng k- order ε-ratonalzable conjecture. Choose any x arg max g N (σ (θ t ) a x θ)t k+ (dθ dt k ) x {0 } T k Then the dfference n expected payoff for t between any (a x) A N and (a x ) under σ n game G N s g N (σ (θ t ) a x θ) g N (σ (θ t ) a x θ) ] t k+ (dθ dt k ) T k = T k T k g N (σ (θ t ) a x θ) g N (σ (θ t ) a x θ) ] t k+ (dθ dt k ) g N (σ (θ t ) a θ) g N(σ (θ t ) a θ) ] t k+ (dθ dt k ) ε = ε hence (a x ) R k+ (t G N ε). Ths provesrk+ (t G N ε) proj A N R k (t G N ε). Conversely, let (a x) R k+ (t G N ε) and let σ : T (A N ) be a correspondng k-order ε-ratonalzable conjecture. Then the dfference n expected payoff for t between any a A N and a under σ n game G N s T k = g N (σ (θ t ) a θ) g N(σ (θ t ) a θ) ] t k+ (dθ dt k ) T k g N (σ (θ t ) a x θ) g N (σ (θ t ) a x θ) ] t k+ (dθ dt k ) ε hence a R k+ (t G N ε). Ths proves proj A N R k (t G N ε) Rk+ (t G N ε), so the proof of () fork + s complete. References Alprants, Charalambos D. and Km C. Border (999), Infnte Dmensonal Analyss, second edton. Sprnger, Berln. 45] Aumann, Robert J. (976), Agreeng to dsagree. Annals of Statstcs, 4, 36 39. 449] Battgall, Perpaolo, Alfredo D Tllo, Edoardo Grllo, and Antono Penta (008), Interactve epstemology and soluton concepts for games wth asymmetrc nformaton. Workng Paper 340, Innocenzo Gasparn Insttute for Economc Research, Boccon Unversty. 446] Brandenburger, Adam and Edde Dekel (993), Herarches of belefs and common knowledge. Journal of Economc Theory, 59, 89 98. 450] Chen, Y-Chun, and Syang Xong (008), Topologes on types: Correcton. Theoretcal Economcs, 3, 83 85. 463]
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